Let G0 be a connected, simply connected real simple Lie group. Suppose that G0 has a compact Cartan subgroup T0, so it has discrete series representations. Relative to T0 there are several distinguished positive root systems + for which there is a unique noncompact simple root ν, the "Borel-de Siebenthal system". There is a lot of fascinating geometry associated to the corresponding "Borel-de Siebenthal discrete series" representations of G0. In this paper we explore some of those geometric aspects and we work out the K0-spectra of the Borel-de Siebenthal discrete series representations. This has already been carried out in detail for the case where the associated symmetric space G0/K0 is of hermitian type, i.e. where ν has coefficient 1 in the maximal root μ, so we assume that the group G0 is not of hermitian type, in other words that ν has coefficient 2 in μ.
